| L(s) = 1 | − 1.73·3-s + 5-s + 3.73·7-s + 3.46·13-s − 1.73·15-s + 2·19-s − 6.46·21-s − 0.535·23-s + 25-s + 5.19·27-s − 8.92·29-s − 5.46·31-s + 3.73·35-s − 4·37-s − 5.99·39-s − 11.9·41-s − 1.73·43-s + 10.6·47-s + 6.92·49-s − 0.535·53-s − 3.46·57-s − 5.46·59-s − 8.46·61-s + 3.46·65-s − 2.26·67-s + 0.928·69-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 0.447·5-s + 1.41·7-s + 0.960·13-s − 0.447·15-s + 0.458·19-s − 1.41·21-s − 0.111·23-s + 0.200·25-s + 1.00·27-s − 1.65·29-s − 0.981·31-s + 0.630·35-s − 0.657·37-s − 0.960·39-s − 1.86·41-s − 0.264·43-s + 1.55·47-s + 0.989·49-s − 0.0736·53-s − 0.458·57-s − 0.711·59-s − 1.08·61-s + 0.429·65-s − 0.277·67-s + 0.111·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 7 | \( 1 - 3.73T + 7T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 0.535T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 0.535T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 + 0.535T + 83T^{2} \) |
| 89 | \( 1 - 2.07T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33475636411160799905922528777, −6.54959536640919787259153116876, −5.67029484988600301239008363935, −5.51823964919837811883385695061, −4.77084556531344421604598035635, −3.97167783720615793765453203007, −3.04810671888920102596975065278, −1.80145002751277339548144609721, −1.35595880952141173863644279068, 0,
1.35595880952141173863644279068, 1.80145002751277339548144609721, 3.04810671888920102596975065278, 3.97167783720615793765453203007, 4.77084556531344421604598035635, 5.51823964919837811883385695061, 5.67029484988600301239008363935, 6.54959536640919787259153116876, 7.33475636411160799905922528777
